Illinois Journal of Mathematics

Minimal Lagrangian submanifolds in indefinite complex space

Henri Anciaux

Full-text: Open access

Abstract

Consider the complex linear space endowed with the canonical pseudo-Hermitian form of arbitrary signature. This yields both a pseudo-Riemannian and a symplectic structure. We prove that those submanifolds which are both Lagrangian and minimal with respect to these structures minimize the volume in their Lagrangian homology class. We also describe several families of minimal Lagrangian submanifolds. In particular, we characterize the minimal Lagrangian surfaces in pseudo-Euclidean complex plane endowed with its natural neutral metric and the equivariant minimal Lagrangian submanifolds of indefinite complex space with arbitrary signature.

Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1331-1343.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1399395835

Digital Object Identifier
doi:10.1215/ijm/1399395835

Mathematical Reviews number (MathSciNet)
MR3231486

Zentralblatt MATH identifier
1290.53074

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index 49Q05: Minimal surfaces [See also 53A10, 58E12]

Citation

Anciaux, Henri. Minimal Lagrangian submanifolds in indefinite complex space. Illinois J. Math. 56 (2012), no. 4, 1331--1343. doi:10.1215/ijm/1399395835. https://projecteuclid.org/euclid.ijm/1399395835


Export citation

References

  • H. Anciaux, B. Guilfoyle and P. Romon, Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface, J. Geom. Phys. 61 (2011), 237–247.
  • H. Anciaux, Minimal submanifolds in pseudo-Riemannian geometry, World Scientific, Hackensack, NJ, 2011.
  • I. Castro and F. Urbano, On a minimal Lagrangian submanifold of ${{\mathbb{C}}^{n}}$ foliated by spheres, Michigan Math. J. 46 (1999), 71–82.
  • B.-Y. Chen, Lagrangian minimal surfaces in Lorentzian complex plane, Arch. Math. (Basel) 91 (2008), no. 4, 366–371.
  • B.-Y. Chen and J.-M. Morvan, Géométrie des surfaces lagrangiennes de ${\mathbb{C}}^{2}$, J. Math. Pures Appl. 66 (1987), 321–325.
  • Y. Dong, On indefinite special Lagrangian submanifolds in indefinite complex Euclidean spaces, J. Geom. Phys. 59 (2009), 710–726.
  • R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47–157.
  • R. Harvey and H. B. Lawson Jr., Split special Lagrangian geometry, Metric and differential geometry: The Jeff Cheeger anniversary volume, Progress in Mathematics, vol. 297, Springer, Basel, 2012, pp. 43–89.
  • R. Harvey, Spinors and calibrations, Academic Press, Boston, MA, 1990.
  • D. Joyce, Constructing special Lagrangian m-folds in $\mathbb{C}^{m}$ by evolving quadrics, Math. Ann. 320 (2001), 757–797.
  • D. Joyce, Y.-I. Lee and M.-P. Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84 (2010), no. 1, 127–161.
  • Y.-I. Lee and M.-T. Wang, Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows, J. Differential Geom. 83 (2009), no. 1, 27–42.
  • J. Mealy, Volume maximization in semi-Riemannian manifolds, Indiana Univ. Math. J. 40 (1991), 793–814.
  • U. Pinkall, Hopf tori in $\mathbb{S}^3$, Invent. Math. 81 (1985), no. 2, 379–386.
  • A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), no. 1–2, 243–259.
  • T. Weinstein, An introduction to Lorentz surfaces, de Gruyter Expositions in Mathematics, vol. 22, de Gruyter, Berlin, 1996.